---
title: "Analyzing proportions with the Arrington et al. 2002 example"
bibliography: "../inst/REFERENCES.bib"
csl: "../inst/apa-6th.csl"
output: 
  rmarkdown::html_vignette
description: >
  This vignette describes how a real dataset with 4 factors can be analyzed.
vignette: >
  %\VignetteIndexEntry{Analyzing proportions with the Arrington et al. 2002 example}
  %\VignetteEngine{knitr::rmarkdown}
  \usepackage[utf8]{inputenc}
---

```{r, echo = FALSE, message = FALSE, results = 'hide', warning = FALSE}
cat("this is hidden; general initializations.\n")
library(ANOPA)

```

@a02 published a data set available from the web.
It presents species of fish and what proportion of them were empty stomached 
when catched. The dataset contained 36000+ catches, which where identified by
their Location (Africa, North America, rest of America), by their Trophism
(their diet, Detritivore, Invertivore, Omnivore, Piscivore) and by the moment
of feeding (Diel: Diurnal or Nocturnal).

The compiled scores can be consulted with
```{r}
ArringtonEtAl2002
```

One first difficulty with this dataset is that some of the cells are missing
(e.g., African fish that are Detrivore during the night). As is the case for 
other sorts of analyses (e.g., ANOVAs), data with missing cells cannot be 
analyzed because the error terms cannot be computed.

One solution adopted by @wh11 was to impute the missing
value. We are not aware if this is an adequate solution, and if so, what imputation
would be acceptable. Consider the following with adequate care.

Warton imputed the missing cells with a very small proportion. In ANOPA, both the 
proportions and the group sizes are required. We implemented a procedure that 
impute a count of 0.05 (fractional counts are not possible from observations, but
are not forbidden in ANOPA) obtained from a single observation.

Consult the default option with 

```{r}
getOption("ANOPA.zeros")
```

The analysis is obtained with

```{r}
w <- anopa( {s; n} ~  Location * Diel * Trophism, ArringtonEtAl2002)
```

The `fyi` message lets you know that cells are missing; the `Warning` message lets you 
know that these cells were imputed (you can suppress messages with 
`options("ANOPA.feedback"="none")`.

To see the result, use `summary(w)` (which shows the corrected and uncorrected statistics)
or `uncorrected(w)` (as the sample is quite large, the correction will be immaterial...), 

```{r}
uncorrected(w)
```

These suggests an interaction Diel : Trophism close to significant.

You can easily make a plot with all the factors using

```{r, message=FALSE, warning=FALSE, fig.width=5, fig.height=5, fig.cap="**Figure 1**. The proportions in the Arrington et al. 2002 data. Error bars show difference-adjusted 95% confidence intervals."}
anopaPlot(w)
```

The missing cells are absent from the plot. To highlight the interaction, restrict the plot to

```{r, message=FALSE, warning=FALSE, fig.width=5, fig.height=3, fig.cap="**Figure 1**. The proportions as a function of class and Difficulty. Error bars show difference-adjusted 95% confidence intervals."}
anopaPlot(w, ~ Trophism * Location)
```

which shows clearly massive difference between Trophism, and small differences between
Omnivorous and Piscivorous fishes with regards to Location.

This can be confirmed by examining simple effects (a.k.a. expected marginal analyzes):

```{r}
e <- emProportions( w, ~ Location * Trophism | Diel  ) 
```

(but it will have to wait for the next version of ANOPA ;-)


# References